Solve for $x$ : $ 4|x - 5| + 5 = 3|x - 5| + 8 $
Explanation: Subtract $ {3|x - 5|} $ from both sides: $ \begin{eqnarray} 4|x - 5| + 5 &=& 3|x - 5| + 8 \\ \\ { - 3|x - 5|} && { - 3|x - 5|} \\ \\ 1|x - 5| + 5 &=& 8 \end{eqnarray} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} 1|x - 5| + 5 &=& 8 \\ \\ { - 5} &=& { - 5} \\ \\ 1|x - 5| &=& 3 \end{eqnarray} $ Simplify: $ |x - 5| = 3$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 5 = -3 $ or $ x - 5 = 3 $ Solve for the solution where $x - 5$ is negative: $ x - 5 = -3 $ Add ${5}$ to both sides: $ \begin{eqnarray} x - 5 &=& -3 \\ \\ {+ 5} && {+ 5} \\ \\ x &=& -3 + 5 \end{eqnarray} $ $ x = 2 $ Then calculate the solution where $x - 5$ is positive: $ x - 5 = 3 $ Add ${5}$ to both sides: $ \begin{eqnarray} x - 5 &=& 3 \\ \\ {+ 5} && {+ 5} \\ \\ x &=& 3 + 5 \end{eqnarray} $ $ x = 8 $ Thus, the correct answer is $x = 2 $ or $x = 8 $.